1. Polynomial Results

2. Polynomial Results
1. If P(x) has k distinct real zeros, a1 , a2 , a3 ,, ak , then;
 x  a1  x  a2  x  a3  x  ak  is a factor of P(x).

3. Polynomial Results
1. If P(x) has k distinct real zeros, a1 , a2 , a3 ,, ak , then;
 x  a1  x  a2  x  a3  x  ak  is a factor of P(x).
e.g. Show that 1 and –2 are zeros of P( x)  x 4  x 3  3 x 2  5 x  10 and
hence factorise P(x).

4. Polynomial Results
1. If P(x) has k distinct real zeros, a1 , a2 , a3 ,, ak , then;
 x  a1  x  a2  x  a3  x  ak  is a factor of P(x).
e.g. Show that 1 and –2 are zeros of P( x)  x 4  x 3  3 x 2  5 x  10 and
hence factorise P(x).
P 1  1  1  3 1  5 1  10
4 3 2
0

5. Polynomial Results
1. If P(x) has k distinct real zeros, a1 , a2 , a3 ,, ak , then;
 x  a1  x  a2  x  a3  x  ak  is a factor of P(x).
e.g. Show that 1 and –2 are zeros of P( x)  x 4  x 3  3 x 2  5 x  10 and
hence factorise P(x).
P 1  1  1  3 1  5 1  10
4 3 2
0
P  2    2    2   3  2   5  2   10
4 3 2
0

6. Polynomial Results
1. If P(x) has k distinct real zeros, a1 , a2 , a3 ,, ak , then;
 x  a1  x  a2  x  a3  x  ak  is a factor of P(x).
e.g. Show that 1 and –2 are zeros of P( x)  x 4  x 3  3 x 2  5 x  10 and
hence factorise P(x).
P 1  1  1  3 1  5 1  10
4 3 2
0
P  2    2    2   3  2   5  2   10
4 3 2
0 1, 2 are zeros of P ( x) and  x  1 x  2  is a factor

7. Polynomial Results
1. If P(x) has k distinct real zeros, a1 , a2 , a3 ,, ak , then;
 x  a1  x  a2  x  a3  x  ak  is a factor of P(x).
e.g. Show that 1 and –2 are zeros of P( x)  x 4  x 3  3 x 2  5 x  10 and
hence factorise P(x).
P 1  1  1  3 1  5 1  10
4 3 2
0
P  2    2    2   3  2   5  2   10
4 3 2
0 1, 2 are zeros of P ( x) and  x  1 x  2  is a factor
P( x)  x 4  x 3  3 x 2  5 x  10

8. Polynomial Results
1. If P(x) has k distinct real zeros, a1 , a2 , a3 ,, ak , then;
 x  a1  x  a2  x  a3  x  ak  is a factor of P(x).
e.g. Show that 1 and –2 are zeros of P( x)  x 4  x 3  3 x 2  5 x  10 and
hence factorise P(x).
P 1  1  1  3 1  5 1  10
4 3 2
0
P  2    2    2   3  2   5  2   10
4 3 2
0 1, 2 are zeros of P ( x) and  x  1 x  2  is a factor
P( x)  x 4  x 3  3 x 2  5 x  10
  x2  x  2  

9. Polynomial Results
1. If P(x) has k distinct real zeros, a1 , a2 , a3 ,, ak , then;
 x  a1  x  a2  x  a3  x  ak  is a factor of P(x).
e.g. Show that 1 and –2 are zeros of P( x)  x 4  x 3  3 x 2  5 x  10 and
hence factorise P(x).
P 1  1  1  3 1  5 1  10
4 3 2
0
P  2    2    2   3  2   5  2   10
4 3 2
0 1, 2 are zeros of P ( x) and  x  1 x  2  is a factor
P( x)  x 4  x 3  3 x 2  5 x  10
  x2  x  2  x2 

10. Polynomial Results
1. If P(x) has k distinct real zeros, a1 , a2 , a3 ,, ak , then;
 x  a1  x  a2  x  a3  x  ak  is a factor of P(x).
e.g. Show that 1 and –2 are zeros of P( x)  x 4  x 3  3 x 2  5 x  10 and
hence factorise P(x).
P 1  1  1  3 1  5 1  10
4 3 2
0
P  2    2    2   3  2   5  2   10
4 3 2
0 1, 2 are zeros of P ( x) and  x  1 x  2  is a factor
P( x)  x 4  x 3  3 x 2  5 x  10
  x2  x  2  x2 5 

11. Polynomial Results
1. If P(x) has k distinct real zeros, a1 , a2 , a3 ,, ak , then;
 x  a1  x  a2  x  a3  x  ak  is a factor of P(x).
e.g. Show that 1 and –2 are zeros of P( x)  x 4  x 3  3 x 2  5 x  10 and
hence factorise P(x).
P 1  1  1  3 1  5 1  10
4 3 2
0
P  2    2    2   3  2   5  2   10
4 3 2
0 1, 2 are zeros of P ( x) and  x  1 x  2  is a factor
P( x)  x 4  x 3  3 x 2  5 x  10
  x2  x  2  x2 5 
  x  1 x  2   x 2  5 

12. 2. If P(x) has degree n and has n distinct real zeros, a1 , a2 , a3 ,, an ,
then P x    x  a1  x  a2  x  a3  x  an 

13. 2. If P(x) has degree n and has n distinct real zeros, a1 , a2 , a3 ,, an ,
then P x    x  a1  x  a2  x  a3  x  an 
3. A polynomial of degree n cannot have more than n distinct real
zeros

14. 2. If P(x) has degree n and has n distinct real zeros, a1 , a2 , a3 ,, an ,
then P x    x  a1  x  a2  x  a3  x  an 
3. A polynomial of degree n cannot have more than n distinct real
zeros
e.g. The polynomial P(x) has a double zero at –7 and a single zero at 2.
Write down;
a) a possible polynomial

15. 2. If P(x) has degree n and has n distinct real zeros, a1 , a2 , a3 ,, an ,
then P x    x  a1  x  a2  x  a3  x  an 
3. A polynomial of degree n cannot have more than n distinct real
zeros
e.g. The polynomial P(x) has a double zero at –7 and a single zero at 2.
Write down;
a) a possible polynomial
P( x)   x  2  x  7 
2

16. 2. If P(x) has degree n and has n distinct real zeros, a1 , a2 , a3 ,, an ,
then P x    x  a1  x  a2  x  a3  x  an 
3. A polynomial of degree n cannot have more than n distinct real
zeros
e.g. The polynomial P(x) has a double zero at –7 and a single zero at 2.
Write down;
a) a possible polynomial
P( x)   x  2  x  7  Q( x)
2
where - Q(x) is a polynomial and does not have a zero at 2 or –7

17. 2. If P(x) has degree n and has n distinct real zeros, a1 , a2 , a3 ,, an ,
then P x    x  a1  x  a2  x  a3  x  an 
3. A polynomial of degree n cannot have more than n distinct real
zeros
e.g. The polynomial P(x) has a double zero at –7 and a single zero at 2.
Write down;
a) a possible polynomial
P( x)  k  x  2  x  7  Q( x)
2
where - Q(x) is a polynomial and does not have a zero at 2 or –7
- k is a real number

18. 2. If P(x) has degree n and has n distinct real zeros, a1 , a2 , a3 ,, an ,
then P x    x  a1  x  a2  x  a3  x  an 
3. A polynomial of degree n cannot have more than n distinct real
zeros
e.g. The polynomial P(x) has a double zero at –7 and a single zero at 2.
Write down;
a) a possible polynomial
P( x)  k  x  2  x  7  Q( x)
2
where - Q(x) is a polynomial and does not have a zero at 2 or –7
- k is a real number
b) a monic polynomial of degree 3.

19. 2. If P(x) has degree n and has n distinct real zeros, a1 , a2 , a3 ,, an ,
then P x    x  a1  x  a2  x  a3  x  an 
3. A polynomial of degree n cannot have more than n distinct real
zeros
e.g. The polynomial P(x) has a double zero at –7 and a single zero at 2.
Write down;
a) a possible polynomial
P( x)  k  x  2  x  7  Q( x)
2
where - Q(x) is a polynomial and does not have a zero at 2 or –7
- k is a real number
b) a monic polynomial of degree 3.
P( x)   x  2  x  7 
2

20. c) A monic polynomial of degree 4

21. c) A monic polynomial of degree 4
P( x)   x  2  x  7 
2

22. c) A monic polynomial of degree 4
P( x)   x  2  x  7   x  a 
2
where a  2 or  7
d) a polynomial of degree 5.

23. c) A monic polynomial of degree 4
P( x)   x  2  x  7   x  a 
2
where a  2 or  7
d) a polynomial of degree 5.
P( x)   x  2  x  7 
2

24. c) A monic polynomial of degree 4
P( x)   x  2  x  7   x  a 
2
where a  2 or  7
d) a polynomial of degree 5.
P( x)   x  2  x  7  Q( x)
2
where - Q(x) is a polynomial of degree 2,
and does not have a zero at 2 or –7

25. c) A monic polynomial of degree 4
P( x)   x  2  x  7   x  a 
2
where a  2 or  7
d) a polynomial of degree 5.
P( x)  k  x  2  x  7  Q( x)
2
where - Q(x) is a polynomial of degree 2,
and does not have a zero at 2 or –7
- k is a real number

26. c) A monic polynomial of degree 4
P( x)   x  2  x  7   x  a 
2
where a  2 or  7
d) a polynomial of degree 5.
P( x)  k  x  2  x  7  Q( x)
2
where - Q(x) is a polynomial of degree 2,
and does not have a zero at 2 or –7
- k is a real number
4. If P(x) has degree n and has more than n real zeros, then P(x) is
the zero polynomial. i.e. P(x) = 0 for all values of x

27. Exercise 4E; 1, 3bd, 4a, 5b, 6a, 8, 12ad